# Titchmarsh-Weyl M-function asymptotics and some results in the inverse spectral theory for vector-valued Sturm-Liouville equations and a certain higher order ordinary differential equation

Research output: ThesisDoctoral Thesis (compilation)

## Abstract

This discourse is constituted by two separate reprots, where the first one offers an elementary deduction of the leading order term asymptotics for the Titchmarsh-Weyl M-function corresponding to a vector-valued Sturm-Liouville equation of the form -(PU')'+QU=zu, xin[0,b), with P^{-1},W,Q being hermitean with locally integrable entries; and under some additional conditions on P^{-1} and W. In the special case of P=W=I, we give some further asymptotic results for the same M-function. In this case, we also prove that the corresponding spectral measure determines the equation uniquely up to conjugation by a constant and unitary matrix R, and we finish this presentation by giving a local form of the Borg-Marchenko theorem in the above case (cf. [GS2, Chapter 3.]); a theorem which is due to Simon, [S], in the scalar case. The object of the second report is to study a higher order ordinary differential equation of the form sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}=zu, xin[0,b), where D=id/dx, and where the coefficients a_{jk}, j,kin[0,m], with a_{mm}=1, satisfy certain regularity conditions and are chosen so that the matrix (a_{jk}) is hermitean. We will also assume that m>1. More precisely, we will prove, using Paley-Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. We will also discuss under which additional conditions the spectral measure uniquely determines the coefficients a_{jk}, j,kin[0,m], j+k<2m, as well as b and the boundary conditions at 0 and (if any) at b.
Original language English Doctor Mathematics (Faculty of Sciences) [unknown], [unknown], Supervisor, External person 2002 Oct 31 Center for Mathematical Sciences, Mathematics, Lund University, Box 118, SE-221 00 LUND, SWEDEN, 91-628-5313-9 Published - 2002

### Bibliographical note

Defence details

Date: 2002-10-31
Time: 10:15
Place: MH:C

External reviewer(s)

Name: W. Desmond Evans, W. Desmond Evans
Title: Prof
Affiliation: Univ. of Wales, Cardiff, UK

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• Mathematics

## Free keywords

• Functions
• differential equations
• Funktioner
• differentialekvationer
• Paley-Wiener.
• The generalized Fourier transform
• A certain higher order ordinary differential equation
• Borg-Marchenko theorems
• Vector-valued Sturm-Liouville equations
• Inverse spectral theory
• Titchmarsh-Weyl M-function asymptotics
• Asymptotics of solutions
• Spectral measure

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